Statistical Analysis of Basketball Shooting in a...

When I watch basketball on television, it is a common occurrence to have an announcer state that some player has the hot-hand. This raises the question: Are Bernoulli trials an adequate model for the outcomes of successive shots in basketball? This paper addresses this question in a controlled (practice) setting. A large simulation study examines the power of the tests that have appeared in the literature as well as tests motivated by the work of Larkey, Smith, and Kadane (LSK). Three test statistics for the null hypothesis of Bernoulli trials have been considered in the literature; one of these, the runs test, is effective at detecting one-step autocorrelation, but poor at detecting nonstationariy. A second test is†¦show more content†¦Their third test is a test of fit and the researchers refer to it as a test of stationarity. The test is nonstandard, but simple to describe. Suppose that the data are 1100100011110101 . . . . Group the data into sets of four, 1100 1000 1111 0101 . . . , and count the number of successes in each set, 2, 1, 4, 2 . . . . Use the 25 counts to test the null hypothesis that the data come from a binomial distribution with n = 4 and p estimated as the proportion of successes obtained in the data. The first difficulty with implementing this test is that typically one or more of the expected counts is quite small. The researchers overcame this problem by combining the O’s and E’s to yield three response categories: fewer than 2, 2, and more than 2, and then applied a χ 2 test with one degree of freedom. The test can be made one-sided by rejecting if and only if the χ 2 test would reject at 0.10 and E O for the middle category (corresponding to two successes). The rationale for this decision rule is that E O in the central category indicates heavier tails, which implies more streakiness. 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